2. An insider's guide to surface curvature and geodesics

Curvature:
If you are a two-dimensional creature living on a smooth surface you
feel its curvature in your guts. When you go from a flat region to one
of positive curvature your insides get stretched with respect to
your periphery; if you enter a region of negative curvature they get
compressed.

For example, suppose you are a disc of radius 1 living on the flat
part of the surface, so your
circumference is 2 = 6.28.. and your area is = 3.14.

Positive:

Suppose part of the surface is curved like the
graph of z= -(x2+y2) (this graph has positive curvature). If you slide over
there so that the middle of
your body is at (0,0,0), your perimeter will fit exactly on the
circle at height -1, but your area will have been stretched to 5.33.. units.

Negative:

On the other hand suppose part of your surface is curved like the graph
of z=x2-y2 (this graph has negative curvature).
If the middle of your body is at the point (0,0,0), your perimeter
will fit the circle in the graph that lies over the circle r=.715..
in the plane; your insides will have to fit in the enclosed area
which is only 2.26.. units.

Geodesics:
The geographers in your two-dimensional universe need to be able
to locate and measure areas of non-zero curvature without risking
their insides. They may have discovered a theorem due to Gauss
which permits these measurements. This theorem is stated in terms
of geodesics. These are the paths on the surface which are
as straight as possible: they turn neither to the left nor to the
right, and their only bending is that which is forced on them by
following the surface.

If the surface is an ordinary plane, the geodesics are ordinary
straight lines.

If the plane is bent without stretching into a cylinder or a cone,
the lines remain geodesics, even though they may bend with the plane.

On a sphere the geodesics are great circles. If you travel along the
equator or along a meridian line, you always move straight ahead. If
you travel along a latitude which is not the equator (think of one
very near the North Pole) you have to keep turning to stay on the
line. Such a latitude is not a geodesic.